Question: Simplify and expand the following expression: $ \dfrac{3}{p - 8}+ \dfrac{4}{p - 1}- \dfrac{5p}{p^2 - 9p + 8} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor the quadratic in the third term: $ \dfrac{5p}{p^2 - 9p + 8} = \dfrac{5p}{(p - 8)(p - 1)}$ Now we have: $ \dfrac{3}{p - 8}+ \dfrac{4}{p - 1}- \dfrac{5p}{(p - 8)(p - 1)} $ The least common multiple of the denominators is: $ (p - 8)(p - 1)$ In order to get the first term over $(p - 8)(p - 1)$ , multiply by $\dfrac{p - 1}{p - 1}$ $ \dfrac{3}{p - 8} \times \dfrac{p - 1}{p - 1} = \dfrac{3(p - 1)}{(p - 8)(p - 1)} $ In order to get the second term over $(p - 8)(p - 1)$ , multiply by $\dfrac{p - 8}{p - 8}$ $ \dfrac{4}{p - 1} \times \dfrac{p - 8}{p - 8} = \dfrac{4(p - 8)}{(p - 8)(p - 1)} $ Now we have: $ \dfrac{3(p - 1)}{(p - 8)(p - 1)} + \dfrac{4(p - 8)}{(p - 8)(p - 1)} - \dfrac{5p}{(p - 8)(p - 1)} $ $ = \dfrac{ 3(p - 1) + 4(p - 8) - 5p} {(p - 8)(p - 1)} $ Expand: $ = \dfrac{3p - 3 + 4p - 32 - 5p}{p^2 - 9p + 8} $ $ = \dfrac{2p - 35}{p^2 - 9p + 8}$